Linear algebra

How to construct a matrix that has column space with vectors: 1 1 0 and 0 1 1 and nullspace with vectors 1 0 1 and 0 0 1?

Any ideas, what do you understand about the terms? It doesnt sound possible to me.

Well, the professor gave us an example when the null space has 1 vector, the example goes like this: Construct a matrix whose column space contains vectors (1, 1, 5) and (0, 3, 1), and whose null space contains the vector (1, 1, 2). This one is easy, obviously the matrix will have 3 rows and 3 columns, the third column has unknown elements a13 a23 and a33. We multiply the matrix with the vector (1,1,2) to get (0,0,0) and the solution is easy. a13= -1/2, a23=-2 and a33=-3. But I don't know how to do it when the null space has 2 vectors.

Agree with the example you were given and your uncertainty about the question. Youve a 3*3 matrix again, so construct the first two columns (column space) then argue about the third one (nullspace).

If youve come across the rank nullity theorem, the argument is straightforward, but its not much more work to try and construct the third column.

I dont understand, how can I multiply the 3x3 matrix (column space) by 3x2 matrix (the nullspace) to get zero vector... it doesn't seem possible.

I dont understand, how can I multiply the 3x3 matrix (column space) by 3x2 matrix (the nullspace) to get zero vector... it doesn't seem possible.

I agree, its not possible. If you try and formulate the equations for the third column values as per your reply, theyd be inconsistent .